## Parallel reductions in $$\lambda$$-calculus.(English)Zbl 0661.03008

The notion of parallel reduction is extracted from the Tait-Martin-Löf proof of the Church-Rosser theorem (for $$\beta$$-reduction). We define parallel $$\beta$$-, $$\eta$$- and $$\beta$$ $$\eta$$-reduction by induction, and use them to give simple proofs of some fundamental theorems in $$\lambda$$- calculus; the normal reduction theorem for $$\beta$$-reduction, that for $$\beta$$ $$\eta$$-reduction, the postponement theorem of $$\eta$$-reduction (in $$\beta$$ $$\eta$$-reduction), and some others.

### MSC:

 03B40 Combinatory logic and lambda calculus

### Keywords:

parallel reduction; $$\lambda$$-calculus
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### References:

 [1] Barendregt, H.P., () [2] Klop, J.W., () [3] Levy, J.J., An algebraic interpretation of the λ-β-K calculus and a labelled λ-calculus, Springer lec. notes comp., 37, 147-165, (1975)
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